Abstract : In this paper, we use an infinite dimensional conditioning function to define a conditional Fourier--Feynman transform (CFFT) and a conditional convolution product (CCP) on the Wiener space. We establish the existences of the CFFT and the CCP for bounded functions which form a Banach algebra. We then provide fundamental relationships between the CFFTs and the CCPs.
Abstract : In this paper, we consider a model problem arising from a classical planar Heisenberg ferromagnetic spin chain \begin{equation*} -\Delta u+V(x)u-\frac{u}{\sqrt{1-u^2}}\Delta \sqrt{1-u^2}=\lambda |u|^{p-2}u,\ x\in\mathbb{R}^{N}, \end{equation*} where $2\leq p<2^*, N\geq 3$. By the Ekeland variational principle, the cut off technique, the change of variables and the $L^{\infty}$ estimate, we study the existence of positive solutions. Here, we construct the $L^{\infty}$ estimate of the solution in an entirely different way. Particularly, all the constants in the expression of this estimate are so well known.
Abstract : Let $\mathcal{P}=\{X_i\colon i\in I\}$ be a partition of a set $X$. We say that a transformation $f\colon X \to X$ preserves $\mathcal{P}$ if for every $X_i \in \mathcal{P}$, there exists $X_j \in \mathcal{P}$ such that $X_if \subseteq X_j$. Consider the semigroup $\mathcal{B}(X,\mathcal{P})$ of all transformations $f$ of $X$ such that $f$ preserves $\mathcal{P}$ and the character (map) $\chi^{(f)}\colon I \to I$ defined by $i\chi^{(f)}=j$ whenever $X_if\subseteq X_j$ is bijective. We describe Green's relations on $\mathcal{B}(X,\mathcal{P})$, and prove that $\mathcal{D} = \mathcal{J}$ on $\mathcal{B}(X,\mathcal{P})$ if $\mathcal{P}$ is finite. We give a necessary and sufficient condition for $\mathcal{D} = \mathcal{J}$ on $\mathcal{B}(X,\mathcal{P})$. We characterize unit-regular elements in $\mathcal{B}(X,\mathcal{P})$, and determine when $\mathcal{B}(X,\mathcal{P})$ is a unit-regular semigroup. We alternatively prove that $\mathcal{B}(X,\mathcal{P})$ is a regular semigroup. We end the paper with a conjecture.
Abstract : In this paper, under some suitable conditions, we study the Spitzer-type law of large numbers for the maximum of partial sums of independent and identically distributed random variables in upper expectation space. Some general results on necessary and sufficient conditions of the Spitzer-type law of large numbers for the maximum of partial sums of independent and identically distributed random variables under sub-linear expectations are established, which extend the corresponding ones in classic probability space to the case of sub-linear expectation space.
Abstract : Let $K$, $H$, $K_{II}$ and $H_{II}$ be the Gaussian curvature, the mean curvature, the second Gaussian curvature and the second mean curvature of a timelike tubular surface $T_\gamma(\alpha)$ with the radius $\gamma$ along a timelike curve $\alpha(s)$ in Minkowski 3-space $E_{1}^3$. We prove that $T_\gamma(\alpha)$ must be a $(K,H)$-Weingarten surface and a $(K,H)$-linear Weingarten surface. We also show that $T_{\gamma}(\alpha)$ is $(X,Y)$-Weingarten type if and only if its central curve is a circle or a helix, where $(X,Y)$ $\in$ $\{(K,K_{II})$, $(K,H_{II})$, $(H,K_{II})$, $(H,H_{II})$, $(K_{II}$, $H_{II}) \}$. Furthermore, we prove that there exist no timelike tubular surfaces of $(X,Y)$-linear Weingarten type, $(X,Y,Z)$-linear Weingarten type and $(K,H,K_{II},H_{II})$-linear Weingarten type along a timelike curve in $E_{1}^3$, where $(X,Y,Z)\in\{(K,H,K_{II})$, $(K,H,H_{II})$, $(K,K_{II},H_{II})$, $(H$, $K_{II},H_{II})\}$.
Abstract : In this paper, the Gauge-Uzawa methods for the Darcy-Brinkman equations driven by temperature and salt concentration \linebreak (DBTC) are proposed. The first order backward difference formula is adopted to approximate the time derivative term, and the linear term is treated implicitly, the nonlinear terms are treated semi-implicit. In each time step, the coupling elliptic problems of velocity, temperature and salt concentration are solved, and then the pressure is solved. The unconditional stability and error estimations of the first order semi-discrete scheme are derived, at the same time, the unconditional stability of the first order fully discrete scheme is obtained. Some numerical experiments verify the theoretical prediction and show the effectiveness of the proposed methods.
Abstract : A vertex coloring of a graph $G$ is called injective if any two vertices with a common neighbor receive distinct colors. A graph $G$ is injectively $k$-choosable if any list $L$ of admissible colors on $V(G)$ of size $k$ allows an injective coloring $\varphi$ such that $\varphi(v)\in L(v)$ whenever $v\in V(G)$. The least $k$ for which $G$ is injectively $k$-choosable is denoted by $\chi_{i}^{l}(G)$. For a planar graph $G$, Bu et al.~proved that $\chi_{i}^{l}(G)\leq\Delta+6$ if girth $g\geq5$ and maximum degree $\Delta(G)\geq8$. In this paper, we improve this result by showing that $\chi_{i}^{l}(G)\leq\Delta+6$ for $g\geq5$ and arbitrary $\Delta(G)$.
Abstract : We solve the Bj\"{o}rling problem for zero mean curvature surfaces in the three-dimensional light cone. As an application, we construct and classify all rotational zero mean curvature surfaces.
Abstract : In this paper, we study nuclearity of semigroup crossed products for quasi-lattice ordered groups. We show the relationships among nuclearity of the semigroup crossed product, amenability of the quasi-lattice ordered group and nuclearity of the underlying $C^*$-algebra.
Abstract : For a finite subgroup $G$ of $GL_n(\mathbb C)$, the moduli space $\mathcal M_{\theta}$ of $\theta$-stable $G$-constellations is rarely smooth. This note shows that for a group $G$ of type $\frac{1}{r}(1,a,b)$ with $r=abc+a+b$, there is a generic stability parameter $\theta\in \Theta$ such that the birational component $Y_{\theta}$ of $\theta$-stable $G$-constellations provides a resolution of the quotient singularity $X:=\mathbb C^3/G$.
Junqiang Zhang
Bull. Korean Math. Soc. 2022; 59(4): 951-960
https://doi.org/10.4134/BKMS.b210562
Yang Liu
Bull. Korean Math. Soc. 2022; 59(5): 1305-1315
https://doi.org/10.4134/BKMS.b210756
Da Woon Jung, Chang Ik Lee, Yang Lee, Sangwon Park, Sung Ju Ryu, Hyo Jin Sung
Bull. Korean Math. Soc. 2022; 59(4): 853-868
https://doi.org/10.4134/BKMS.b210495
Bikash Chakraborty
Bull. Korean Math. Soc. 2022; 59(5): 1247-1253
https://doi.org/10.4134/BKMS.b210700
Namjip Koo, Hyunhee Lee, Nyamdavaa Tsegmid
Bull. Korean Math. Soc. 2024; 61(1): 195-205
https://doi.org/10.4134/BKMS.b230071
Yong Lin, Yuanyuan Xie
Bull. Korean Math. Soc. 2022; 59(3): 745-756
https://doi.org/10.4134/BKMS.b210445
Gholamreza Pirmohammadi
Bull. Korean Math. Soc. 2024; 61(1): 273-280
https://doi.org/10.4134/BKMS.b230110
Kui Hu, Hwankoo Kim, Dechuan Zhou
Bull. Korean Math. Soc. 2022; 59(5): 1317-1325
https://doi.org/10.4134/BKMS.b210759
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